Theorem onexlimgt 43689

Description: For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)

Assertion

Ref	Expression
onexlimgt	⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem onexlimgt

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 9558	. . . 4 ⊢ ω ∈ On
2		onun2 6427	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ∪ ω) ∈ On)
3	1, 2	mpan2 692	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∪ ω) ∈ On)
4		onexomgt 43687	. . 3 ⊢ ((𝐴 ∪ ω) ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
5	3, 4	syl 17	. 2 ⊢ (𝐴 ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
6		simp2 1138	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝑎 ∈ On)
7		omcl 8464	. . . . 5 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
8	1, 6, 7	sylancr 588	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (ω ·o 𝑎) ∈ On)
9		noel 4279	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∪ ω) ∈ ∅
10		oveq2 7368	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = (ω ·o ∅))
11		om0 8445	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → (ω ·o ∅) = ∅)
12	1, 11	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (ω ·o ∅) = ∅
13	10, 12	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = ∅)
14	13	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ (𝐴 ∪ ω) ∈ ∅))
15	14	notbid 318	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
16	15	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
17	9, 16	mpbiri 258	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
18	17	pm2.21d 121	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎)))
19	18	ex 412	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎))))
20	19	com23 86	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → (𝑎 = ∅ → Lim (ω ·o 𝑎))))
21	20	3impia 1118	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ → Lim (ω ·o 𝑎)))
22		limom 7826	. . . . . . . . 9 ⊢ Lim ω
23	1, 22	pm3.2i 470	. . . . . . . 8 ⊢ (ω ∈ On ∧ Lim ω)
24	6, 23	jctir 520	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)))
25		omlimcl2 43688	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
26	24, 25	sylan 581	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
27	26	ex 412	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (∅ ∈ 𝑎 → Lim (ω ·o 𝑎)))
28		on0eqel 6442	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
29	6, 28	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
30	21, 27, 29	mpjaod 861	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → Lim (ω ·o 𝑎))
31		simp1 1137	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ On)
32	31, 8	jca 511	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On))
33		simp3 1139	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
34		ssun1 4119	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ ω)
35	33, 34	jctil 519	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)))
36		ontr2 6365	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On) → ((𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎)))
37	32, 35, 36	sylc 65	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎))
38		limeq 6329	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (Lim 𝑥 ↔ Lim (ω ·o 𝑎)))
39		eleq2 2826	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (ω ·o 𝑎)))
40	38, 39	anbi12d 633	. . . . 5 ⊢ (𝑥 = (ω ·o 𝑎) → ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ↔ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))))
41	40	rspcev 3565	. . . 4 ⊢ (((ω ·o 𝑎) ∈ On ∧ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
42	8, 30, 37, 41	syl12anc 837	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
43	42	rexlimdv3a 3143	. 2 ⊢ (𝐴 ∈ On → (∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥)))
44	5, 43	mpd 15	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  Oncon0 6317  Lim wlim 6318  (class class class)co 7360  ωcom 7810   ·_o comu 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-oadd 8402  df-omul 8403

This theorem is referenced by: (None)